= Euler discretization of logistic growth
{c}
{title2=$\lambda=1+r\Delta t$}
The <Forward Euler method> applied to the <logistic differential equation> $y'=ry(1-ay)$ becomes the <logistic map> after $u=a(\lambda-1)y/\lambda$, $\lambda=1+r\Delta t$. The positive <equilibrium point> of the <differential equation> has perturbations decaying as $e^{-rt}$. Its discrete <fixed-point multiplier> is $1-r\Delta t=2-\lambda$: for $r\Delta t>2$, the discretization has an artificial oscillatory <instability>, leading initially to alternation on successive steps. This contrasts with the positive perturbation multiplier $e^{-r\Delta t}$ of the exact time-step map.
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